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1.56 Example

The integral $\int_{0}^{1}{{dx}\over{x(1+x^{2})}}$ diverges.

Solution. We simplify the integral using partial fractions:

{\frac{1}{x(1+x^{2})}=\frac{A}{x}+\frac{Bx+C}{x^{2}+1}}

where $A(x^{2}+1)+(Bx+C)x=1$ , so that $A=1$ , $B=-1$ and $C=0$ . Thus

\int_{\delta}^{1}\frac{dx}{x(1+x^{2})}={\int_{\delta}^{1}\left(\frac{1}{x}-% \frac{x}{1+x^{2}}\right)dx}{\,=\left[\log x-\frac{1}{2}\log(1+x^{2})\right]_{% \delta}^{1}}

{=-\frac{1}{2}\log 2-\log\delta+\frac{1}{2}\log(1+\delta^{2}).}

Since every term except $\log\delta$ converges as $\delta\rightarrow 0$ , the integral diverges.